The three-dimensional Lieb–Thirring constant
For a real-valued potential $V \in L^{5/2}(\mathbb{R}^3)$, let $H = -\Delta + V$ on $L^2(\mathbb{R}^3)$ be defined by its quadratic form, let $E_1 \le E_2 \le \cdots < 0$ be its negative eigenvalues, counted with multiplicity, and let $V_-(x) = \max\{-V(x), 0\}$. Lieb and Thirring proved $$\sum_j |E_j| \le L_{1,3} \int_{\mathbb{R}^3} V_-(x)^{5/2}\, dx$$ with a constant $L_{1,3}$ independent of $V$. Weyl asymptotics for $-\Delta + \lambda V$ as $\lambda \to \infty$ show that the best constant is at least the semiclassical constant $L^{\mathrm{cl}}_{1,3} = 1/(15\pi^2)$. Laptev and Weidl proved that the semiclassical constants are sharp for the moments $\sum_j |E_j|^\gamma$ with $\gamma \ge 3/2$ in every dimension, while for $\gamma < 1$ they are not sharp in dimensions $d \ge 2$ (Helffer and Robert; see Schimmer). Frank, Hundertmark, Jex and Nam proved the inequality with constant $1.456\, L^{\mathrm{cl}}_{1,3}$, and Read and Schulz (2026) with $(2/\sqrt{3})\, L^{\mathrm{cl}}_{1,3}$.
Problem. Is $$\sum_j |E_j| \le \frac{1}{15\pi^2} \int_{\mathbb{R}^3} V_-(x)^{5/2}\, dx$$ for every real-valued $V \in L^{5/2}(\mathbb{R}^3)$?
Source: Lieb and Thirring (1976), the case d = 3, γ = 1 of their conjecture on the sharp constants; see Schimmer (2022).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Lieb–Thirring inequalities, Schrödinger operators, sharp constants
References
@incollection{lieb1976,
author = {Lieb, Elliott H. and Thirring, Walter E.},
title = {Inequalities for the moments of the eigenvalues of the {S}chr{\"o}dinger {H}amiltonian and their relation to {S}obolev inequalities},
booktitle = {Studies in Mathematical Physics: Essays in Honor of {V}alentine {B}argmann},
editor = {Lieb, Elliott H. and Simon, Barry and Wightman, Arthur S.},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1976},
pages = {269--303},
doi = {10.1515/9781400868940-014},
}
@incollection{schimmer2022,
author = {Schimmer, Lukas},
title = {The state of the {L}ieb--{T}hirring conjecture},
booktitle = {The Physics and Mathematics of {E}lliott {L}ieb. {T}he 90th Anniversary. {V}olume {II}},
editor = {Frank, Rupert L. and Laptev, Ari and Lewin, Mathieu and Seiringer, Robert},
publisher = {EMS Press},
address = {Berlin},
year = {2022},
pages = {253--275},
doi = {10.4171/90-2/39},
eprint = {2203.06051},
archivePrefix = {arXiv},
}
@article{laptev2000,
author = {Laptev, Ari and Weidl, Timo},
title = {Sharp {L}ieb--{T}hirring inequalities in high dimensions},
journal = {Acta Math.},
volume = {184},
number = {1},
year = {2000},
pages = {87--111},
doi = {10.1007/BF02392782},
}
@article{frank2021,
author = {Frank, Rupert L. and Hundertmark, Dirk and Jex, Michal and Nam, Phan Th{\`a}nh},
title = {The {L}ieb--{T}hirring inequality revisited},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {23},
number = {8},
year = {2021},
pages = {2583--2600},
doi = {10.4171/JEMS/1062},
eprint = {1808.09017},
archivePrefix = {arXiv},
}
@misc{read2026,
author = {Read, Larry and Schulz, Marvin R.},
title = {The sharp one-dimensional {L}ieb--{T}hirring inequality for the sum of eigenvalues},
year = {2026},
eprint = {2609.10478},
archivePrefix = {arXiv},
}
@article{helffer1990,
author = {Helffer, Bernard and Robert, Didier},
title = {Riesz means of bounded states and semi-classical limit connected with a {L}ieb--{T}hirring conjecture. {II}},
journal = {Ann. Inst. H. Poincar{\'e} Phys. Th{\'e}or.},
volume = {53},
number = {2},
year = {1990},
pages = {139--147},
}
@misc{aim-lieb-thirring-three-dimensional,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The three-dimensional Lieb–Thirring constant}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/018-lieb-thirring-three-dimensional.md},
note = {Entry 018 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00292, "The three-dimensional Lieb–Thirring constant", clopen.solutions.